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Finding a Linear Function for any Two Points

Module by: Kenny M. Felder. E-mail the author

Summary: This module discusses the concept of modeling data with linear functions in Algebra.

Finding a Linear Function for any Two Points

In an earlier unit, we did a great deal of work with the equation for the height of a ball thrown straight up into the air. Now, suppose you want an equation for the speed of such a ball. Not knowing the correct formula, you run an experiment, and you measure the two data points.

Table 1
t (time) v (velocity, or speed)
1 second 50 ft/sec
3 seconds 18 ft/sec

Obviously, the ball is slowing down as it travels upward. Based on these two data points, what function v ( t ) v(t) might model the speed of the ball?

Given any two points, the simplest equation is always a line. We have two points, (1,50) and (3,18). How do we find the equation for that line? Recall that every line can be written in the form:

y = m x + b y=mx+b
(1)

If we can find the m and b for our particular line, we will have the formula.

Here is the key: if our line contains the point (1,50) that means that when we plug in the x-value 1, we must get the y-value 50.

The derivation of the equation of the line given the ordered pair and slope when m = 1.

Similarly, we can use the point (3,18) to generate the equation 18 = m ( 3 ) + b 18=m(3)+b. So now, in order to find m m and b b, we simply have to solve two equations and two unknowns! We can solve them either by substitution or elimination: the example below uses substitution.

m + b = 50 b = 50 - m m+b=50b=50-m
(2)
3 m + b = 18 3 m + ( 50 - m ) = 18 3m+b=183m+(50-m)=18
(3)
2 m + 50 = 18 2m+50=18
(4)
2 m = 32 2m=32
(5)
m = - 16 m=-16
(6)
b = 50 - ( - 16 ) = 66 b=50-(-16)=66
(7)

So we have found m m and b b. Since these are the unknowns the in the equation y = m x + b y=mx+b, the equation we are looking for is:

y = 16 x + 66 y=16x+66
(8)

Based on this equation, we would expect, for instance, that after 4 seconds, the speed would be 2 ft/sec. If we measured the speed after 4 seconds and found this result, we would gain confidence that our formula is correct.

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